Understanding 2D and 3D Elastic Collisions: Solving Analytical Problems

[Omri]

Hello,

I have recently been interested in the problem of 2- and 3-dimensional elastic collisions. I just don't understand how to solve these problems analytically: in the 2D case we have 4 variables (x,y components of the velocity times 2 bodies) and only 3 equations (2 conservation of momentum, 1 conservation of energy); in the 3D case (similarly) we have 6 variables and only 4 equtions.
I ran across this page: http://www.plasmaphysics.org.uk/collision2d.htm
but I stopped understanding when they started talking about theta as the sum of two other angles.
I would be happy if somebody could explain it to me.

Thanks a lot!

 

[AlephZero]

In the 2D case, the "fourth equation" comes from the fact that you know the direction of the momentum change, from the geometry of the collision.

Resolve the velocities along the line of impact and tangential to it. The two tangential velocity components don't change, because there is no impact force in the tangential direction.

Apply conservation of momentum and energy along the line of impact: that gives two equations to find the other two velocity components.

In 3D there there are no velocity changes in the plane tangent to the impact, so 4 components of velocity don't change. Again, the two conservation equations give the two velocities along the line of impact.

 

[Omri]

I more or less get the idea, but the equations in the webpage (again, starting from the weird angles equations) sort of confused me.
Could you please explain what happened there mathematically (I'm referring to the 2D case)?